COVARIANCE, COVARIANCE AND VARIANCE RULES, AND CORRELATION
Covariance
cov( X , Y ) XY E ( X X )(Y Y )
The covariance of two random variables X and Y, often written XY, is defined to be the
expected value of the product of their deviations from their popul
PRECISION OF THE REGRESSION COEFFICIENTS
Simple regression model:
Y = 1 + 2X + u
probability density
function of b2
2
b2
We have seen that the regression coefficients b1 and b2 are random variables. They provide
point estimates of 1 and 2, respectively. I
Cost Analysis
The meaning and measurement of cost
Short-run Cost Functions
Long-run Cost Functions
Scale Economies and Cost
Cobb-Douglas & Long Run
Cost
Slide 1
The Object of Cost Analysis
Managers seek to produce the highest quality
products at the low
Topic
Budgetary and
Other Constraints on Choice
N V M Rao
Consumption Choice Sets
A consumption choice set is the
collection of all consumption
choices available to the consumer.
What constrains consumption
choice?
Budgetary, time and other
resource limi
TOPIC
Choice
Economic Rationality
The principal behavioral postulate is
that a decisionmaker chooses its
most preferred alternative from those
available to it.
The available choices constitute the
choice set.
How is the most preferred bundle in
the choice
Chapter 12
The Partial
Equilibrium
Competitive Model
Market Demand
Assume that there are only two goods (x and
y)
An individuals demand for x is
n
Quantity of x demanded = x(px,py,I)
Market demand for X x i ( px , py , I i )
i
1
If we use i to reflect eac
SIMPLE REGRESSION MODEL
Y
Y 1 2 X
1
X1
X2
X3
X4
X
Suppose that a variable Y is a linear function of another variable X, with unknown
parameters 1 and 2 that we wish to estimate.
1
SIMPLE REGRESSION MODEL
Y
Y 1 2 X
1
X1
X2
X3
X4
X
Suppose that we have a sa
GAUSS-MARKOV CONDITIONS
&
GAUSS-MARKOV THEOREM
GAUSS-MARKOV CONDITIONS AND UNBIASEDNESS OF b1 AND b2
Simple regression model: Y = 1 + 2X + u
Gauss-Markov conditions
1.
E(ui) = 0
This sequence will demonstrate that the OLS estimators of the regression coef
1
TOPIC
Review
Some Basic
Statistical Concepts
Random Variable
random variable:
A variable whose value is unknown until it is observed.
The value of a random variable results from an experiment.
The term random variable implies the existence of some
known
UNBIASEDNESS AND EFFICIENCY
Unbiasedness of
X:
1
1
E ( X ) E ( X 1 . X n ) E ( X 1 . X n )
n
n
1
1
E ( X 1 ) . E ( X n ) n X X
n
n
Much of the analysis in this course will be concerned with three properties of estimators:
unbiasedness, efficiency, and
TESTING A HYPOTHESIS RELATING TO THE POPULATION MEAN
Assumption:
X ~ N( , 2)
Null hypothesis:
H 0 : 0
Alternative hypothesis:
H 1 : 0
This sequence describes the testing of a hypothesis at the 5% and 1% significance levels.
It also defines what is meant b
CONFIDENCE INTERVALS
probability density function of X
conditional on = 0 being true
null hypothesis
H0: = 0
2.5%
01.96sd 0sd
2.5%
0
0+sd 0+1.96sd
In the sequence on hypothesis testing, we started with a given hypothesis, for example H0:
= 0, and consi
GOODNESS OF FIT
Four useful results:
e 0
Y Y
X e
i i
0
Y e
i i
0
This sequence explains measures of goodness of fit in regression analysis. It is
convenient to start by demonstrating four useful results. The first is that the mean value of
the residuals m
F TEST OF GOODNESS OF FIT
(Y Y ) 2 (Y Y )2 e 2
TSS ESS RSS
In an earlier sequence it was demonstrated that the sum of the squares of the actual values
of Y (TSS: total sum of squares) could be decomposed into the sum of the squares of the
fitted values (E
1
Topic:
Applications of Dummy Independent Variables
The Normality Assumption
In general we assume the error term is
normally distributed.
Financial data often fails this assumption due
to the volatile nature of the data and the
numbers of outliers.
Th
t TEST OF A HYPOTHESIS RELATING TO A REGRESSION COEFFICIENT
s.d. of b2 known
discrepancy between
hypothetical value and sample
estimate, in terms of s.d.:
0
b2 2
z
s.d.
5% significance test:
reject H0: 2 = 20 if
z
> 1.96 or
z
< -1.96
The diagram summarize
MULTICOLLINEARITY
WHAT HAPPENS IF THE REGRESSORS ARE
CORRELATED?
MULTICOLLINEARITY
What is the nature of multicollinearity?
Is multicollinearity really a problem?
What are its practical consequences?
How does one detect it?
What remedial measures can be t